محاضرة عناصر البحث والاحصاء

To compute the expected frequency, fe, in any cell, you need to understand that if the null
hypothesis is true, the proportion of successes in the two populations will be equal. Then the
sample proportions you compute from each of the two groups would differ from each other
only by chance. Each would provide an estimate of the common population parameter, ?. A statistic
that combines these two separate estimates together into one overall estimate of the population
parameter provides more information than either of the two separate estimates could
provide by itself. This statistic, given by the symbol represents the estimated overall proportion
of successes for the two groups combined (that is, the total number of successes divided by
the total sample size). The complement of , 1 ? , represents the estimated overall proportion
of failures in the two groups. Using the notation presented in Table 12.1 on page 463,
Equation (12.2) defines .
COMPUTING THE ESTIMATED OVERALL PROPORTION
(12.2)
To compute the expected frequency, fe, for each cell pertaining to success (that is, the cells
in the first row in the contingency table), you multiply the sample size (or column total) for a
group by . To compute the expected frequency, fe, for each cell pertaining to failure (that is,
the cells in the second row in the contingency table), you multiply the sample size (or column
total) for a group by (1 ? ).
The test statistic shown in Equation (12.1) approximately follows a chi-square distribution
(see Table E.4) with 1 degree of freedom. Using a level of significance ?, you reject the
null hypothesis if the computed ?2 test statistic is greater than , the upper-tail critical value
from the ?2 distribution having 1 degree of freedom. Thus, the decision rule is
Figure 12.1 illustrates the decision rule.
Reject if
otherwise, do not reject
H0
H
? ?U 2 2
0
> ;
.
?U 2
p
p
p
X X
n n
X
n
= +
+
1 2 =
1 2
p